Appendix 19.1: CAPM

CAPM as a Basis for Deriving IRR, PME, and Risk-Adjusted Returns

I am going to derive all the performance measures from the basic CAPM relationship.

Basic CAPM:

Assume risk-free rate is zero for now and rewrite the asset's return as:

Solve this for P₀ to get the pricing form of the CAPM:

This is a simple two-period discount model. Generalize the future price as a stream of cash flows (dividends) in discrete time.

If we set t = 1 and assume the market rate is constant at r_m, then this looks like the simple textbook dividend discount model, but with a CAPM discount factor, that is a function of the asset's β with the market return, plus Jensen's index (α).

The discount factor is the risk-adjusted market rate of return plus the alpha. Equivalently, we can express this as a future value by multiplying both sides by (1 + α + βr_m)^T to get:

Finally, we can rewrite this in PME form as:

The terms in the parentheses are the discount factors that make the risk-adjusted value of the investment (P₀) just equal to the sum of the risk-adjusted value of the dividends.

GMM is an econometric method that estimates the values of the CAPM parameters α and β that, in conjunction with observed market returns (the opportunity cost of capital), equate the risk-adjusted present value of the investment to its return over time.

Other performance measures such as IRR, PME, and Long and Nickels alpha are all extensions of this basic relationship but with restrictions on the values of α and β. These performance measures must come at a cost and therefore, in principle, are inferior to the CAPM. The costs materialize because they impose specific restrictions on the level of risk (β) and abnormal return (α) that are not necessarily consistent with the underlying cash flows given the market returns for the investment period.

Let's now generalize the basic CAPM relationship:

We get the following set of relationships with their required restrictions:

Case 1: Multiple (α = β = 0)

Case 2: PME (α = 0, β = 1)

Case 3: IRR (m = 1, β = 0, solve α)

Case 4: GMM (m = 1, solve α, β)